On principal points for location mixtures of spherically symmetric distributions

In this paper, we investigate some properties of 2-principal points for location mixtures of spherically symmetric distributions with focus on a linear subspace in which a set of 2-principal points must lie. Our results can be viewed as an extension of those of Yamamoto and Shinozaki 2000. Two principal points for multivariate location mixtures of spherically symmetric distributions. J. Japan Statist. Soc. 30, 53–63], where a finite location mixture of spherically symmetric distributions is treated. As an extension of their paper, this paper defines a wider class of distributions, and derives a linear subspace in which a set of 2-principal points must exist. A theorem useful for comparing the mean squared distances is also established.